The mean field game limit in the present context, where interaction is also through the control, was first proved by [1]}.
In this setting we should also mention results by [2]} for games in the weak formulation and [3]} using relaxed controls.
In the nutshell, it was proved in [1]} that when the coefficients of the game are sufficiently regular (see the proper statement below) and the \(N\) -player game admits at least one Nash equilibrium, then this Nash equilibrium converges in the \(\mathbb {L}^2\) –sense to a mean field equilibrium.
The aim of the present work is to make a finer analysis of this convergence result by deriving large deviation principles both for the Nash equilibrium and the state processes of the players at equilibrium.
We will recall the concept of large deviation principle (LDP) in Section ; but note already that this allows to quantify the probability that the Nash equilibrium does not converge to the mean field equilibrium, a rare event in view of the results in [1]}.
Our main results (stated in the next section) state that if in addition to the convergence assumptions of [1]} we assume that a certain (master) equation admits a Lipschitz continuous solutions, then the Nash equilibrium satisfies the LDP.
In contrast to [7]}, our model includes interaction of the players through their controls, but also deals with open-loop equilibria.
Moreover, the assumptions of the our second LDP result (Theorem REF ) do not require or a priori guarantee uniqueness of the mean field equilibrium.
In addition, we propose a different proof, one that is based on forward backward stochastic differential equations (FBSDE) and their decoupling fields.
We will elaborate more on the idea of proof in the discussion following the statement of Theorem REF .
Note that this method seems versatile enough to be applied to the LDP of cooperative games.
